With the possible exception of apostrophe, none of those examples is necessarily particularly possessive, but context can make all the difference.
Unlike, say, a school's mascot, which is uniquely associated as being owned by the school (as a sort of branding), a school project is typically something that individuals do as a part of their school work. In this case, school is an adjective describing the general type of project, not, specifically, the owner of the project. An exception here might be a unified school-sponsored event; the school's (annual service) project might be to clean litter off campus, for example, but that's kind of a contrived example.
Similarly, the service is something that's done generally to the car, it's not something that would be uniquely associated as being possessed specifically by the car. For example, the car service may be done to any number of cars, but that car's bumper (as a possessive counter-example) might be difficult to replace. I think, again, depending on context, car's service could be legitimate. For example, my friend provides a standard car service to people around town, but my car's service might include a little extra detailing.
The apostrophe is a bit trickier, I think, because in this case, the particular usage is unique to apostrophes. I might be (probably am) wrong, but I don't see anything incorrect with referring to apostrophes' usage, although I think it's more common to refer to the non-possessive apostrophe usage. That may be because usage implies a passivity on the part of the apostrophe (which is inanimate to begin with), so it's not considered something that the apostrophes own a specific claim to, but rather simply something that is done, in general, to apostrophes.
In all those cases, the possessives of the plurals which end with s would be a simple apostrophe. (e.g. cars', not cars's).